Coherent charge and spin oscillations induced by local quenches in nanowires with spin-orbit coupling

When a local and attractive potential is quenched in a nanowire, the spectrum changes its topology from a purely continuum to a continuum and discrete portion. We show that, under appropriate conditions, this quench leads to stable coherent oscillations in the observables' time evolution. In particular, we demonstrate that ballistic nanowires with spin-orbit coupling (SOC) exposed to a uniform magnetic field are especially suitable to observe this effect. Indeed, while in ordinary nanowires the effect occurs only if the strength $U_0$ of the attractive potential is sufficiently strong, even a weak value of $U_0$ is sufficient in SOC nanowires. Furthermore, in these systems coherent oscillations in the spin sector can be generated and controlled electrically by quenching the gate voltage acting on the charge sector. We interpret the origin of this phenomenon, analyze the effect of variation of the chemical potential and the switching time of the quenched attractive potential, and address possible implementation schemes.

Figure 1

Sketch of the setup. A SOC nanowire coupled to narrow metallic gates, which generate a localized potential. An attractive gate potential is quenched at the time $t=0$, generating localized bound states near the gated region, and causing coherent oscillations both in the charge and in the spin sector of the nanowire observables.

Figure 2

The case of a conventional NW after the quench of an attractive Gaussian potential with ${\lambda }_U=150\mathrm{nm},\mu =0.2\mathrm{meV}$, and three different values of the potential strength: (a) $U_0=0.15\mathrm{meV}$, (b) $U_0=0.30\mathrm{meV}$, and (c) $U_0=0.60\mathrm{meV}$. The left panels show the corresponding number of localized bound states, the color denoting the parity (blue for $p=+1$ and red for $p=-1)$. The parabolic band of the pre-quench Hamiltonian is shown as a guide to the eye. The right panels display the corresponding behavior of the electron density at the origin (center of the quenched potential) normalized to its pre-quench value ${\rho }_{\mathrm{pre}}$; the left panels depict schematically the energy of the bound states per spin band and the parity $p_n$ (blue for $+1$, cyan for $-1$).

Figure 3

The time evolution of the charge density $\rho$ at $x=0$ (normalized to the constant and uniform pre-quench value ${\rho }_{\mathrm{pre}})$ after a sudden quench of the attractive Gaussian potential, in the case of a SOC nanowire with $E_{\mathrm{SO}}=0.05\mathrm{meV}$ (thin red solid curve) and $E_{\mathrm{SO}}=0.25\mathrm{meV}$ (thick black solid curve); other parameters given in Sec. 4a. Coherent oscillations are clearly visible. For comparison, the case of a conventional nanowire with the same parameters, except for vanishing SOC, is also plotted as shown (dashed curve). In that case oscillations rapidly decay.

Figure 4

(a) Scheme of the four bound states of the post-quench Hamiltonian $H^{\prime }(t>0)$ as found by exact diagonalization. For reference, we plot also the energy bands of the nanowire before the quench. Parameters as in Sec. 4a. (b) Plot of the density profile of the four bound states.

Figure 5

Time evolution of the spin polarization components after a sudden quench of the attractive Gaussian potential (parameter values given in Sec. 4a): (a) the $P_x$ component at the origin $x=0$; (b) and (c) the $P_y$ and $P_z$ components at $x=+200\mathrm{nm}$. Solid curves refer to the case of a SOC nanowire, whose polarization components exhibit coherent oscillations. For comparison, dashed curves refer to the case of a conventional nanowire: Only the component $P_x$ along the direction of the applied magnetic field is nonvanishing (and is constant in time), whereas the two components $P_y$ and $P_z$ orthogonal to the applied field are vanishing.

Figure 6

Space-time colormap plots of the three components of the spin polarization given in Eq. (22) after a sudden quench of the impurity along (a) the $x$ direction; (b) the $y$ direction; and (c) the $z$ direction. Parameters are given in Sec. 4a.

Figure 7

The interplay between the SOC and the external magnetic field gives rise to an effective inhomogeneous magnetic field ${\mathbf{h}}_{\mathrm{eff}}\left(x\right)$ [see Eq. (25)], whose direction rotates over the spin-orbit length $l_{\mathrm{SO}}$, whereas the attractive quench potential extends over a lengthscale ${\lambda }_U$ around the origin, which also corresponds to the typical lengthscale of the bound states. (a) In the regime ${\lambda }_U>l_{\mathrm{SO}}$ electron spin in the bound states experience the inhomogeneities of ${\mathbf{h}}_{\mathrm{eff}}\left(x\right)$ and rapidly varies along the bound states, causing a finite overlap between different bound states and thereby leading the coherent oscillations to appear after the quench. (b) In the opposite regime ${\lambda }_U<l_{\mathrm{SO}}$ the electron spin cannot experience the long-wavelength inhomogeneity of the effective field, so that bound states are spin orthogonal: The spin selection causes the decay of the density. In particular, the limit of $l_{\mathrm{SO}}\to \infty$ corresponds to the case of conventional NWs.

Figure 8

Time evolution of the spin polarization component $P_x\left(t\right)$ (along the external applied magnetic field) after the quench, for different values of the chemical potential $\mu$ (all other parameters are as in Sec. 4a). To help locate the various chemical potential values with respect to the pre-quench energy band extrema and to the post-quench bound state energies—shown in Fig. 4—they have also been reported here on the right.

Figure 9

(a) The time evolution of the charge density after a quench performed with different switching time ${\tau }_{sw}$ of the ramp protocol function (26). While the thick black curve corresponds to the case of sudden quench $\left({\tau }_{sw}\to 0\right)$ analyzed in the previous section, the dashed red curve and the thin blue curve describe the behavior for ${\tau }_{sw}$ comparable and much longer than the coherent oscillations period $\approx 8\mathrm{ps}$, respectively. (b) The same as panel (a) for the $P_x$ component of the spin polarization.